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Attribute-driven edge bundling for general graphs withapplications in trail analysis

Vsevolod Peysakhovich, Christophe Hurter, Alexandru C Telea

To cite this version:Vsevolod Peysakhovich, Christophe Hurter, Alexandru C Telea. Attribute-driven edge bundling forgeneral graphs with applications in trail analysis. The 8th IEEE Pacific Visualization Symposium(PacificVis 2015), Apr 2015, Hangzhou, China. pp. 39-46, 10.1109/PACIFICVIS.2015.7156354.hal-01411568

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This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 16557

To cite this version: Peysakhovich, Vsevolod and Hurter, Christophe and Telea, Alexandru Attribute-driven edge bundling for general graphs with applications in trail analysis. (2015) In: The 8th IEEE Pacific Visualization Symposium (PacificVis 2015), 14 April 2015 - 17 April 2015 (Hangzhou, China).

To link this article: http://dx.doi.org/10.1109/PACIFICVIS.2015.7156354

Attribute-Driven Edge Bundling for General Graphswith Applications in Trail Analysis

Vsevolod Peysakhovich∗

ISAE, Toulouse, FranceChristophe Hurter†

DGAC, Toulouse, FranceAlexandru Telea‡

University of Groningen,Netherlands

ABSTRACT

Edge bundling methods reduce visual clutter of dense and occludedgraphs. However, existing bundling techniques either ignore edgeproperties such as direction and data attributes, or are otherwise com-putationally not scalable, which makes them unsuitable for taskssuch as exploration of large trajectory datasets. We present a newframework to generate bundled graph layouts according to any nu-merical edge attributes such as directions, timestamps or weights.We propose a GPU-based implementation linear in number of edges,which makes our algorithm applicable to large datasets. We demon-strate our method with applications in the analysis of aircraft trajec-tory datasets and eye-movement traces.

Index Terms: I.3.3 [Computing Methodologies]: ComputerGraphics—Picture/Image Generation; I.3.6 [Computing Methodolo-gies]: Computer Graphics—Methodology and Techniques

1 INTRODUCTION

Large attributed graphs are ubiquitous in many application domains,such as traffic analysis and planning, network analysis, and bioinfor-matics. They represent a special subclass of general graphs whereedge directions encode essential information for the analysis task athand. For such graphs, classical visual metaphors such as node-linkdiagrams produce too much clutter to generate useful pictures.

Graph bundling methods attempt to reduce clutter by groupingedges found to be compatible into so-called bundles. Yet, few suchmethods handle attributed graphs – edge compatibility is mainlybased on spatial position and does not use attributes such as edge di-rection or data values. Reasoning about such attributes (seeing theirvalues reflected in the bundling) is essential in many applications.Bundling methods for attributed graphs exist, but are computation-ally not scalable, and thus not usable for large-scale data exploration.

We present Attribute-Driven Edge Bundling (ADEB), a general-ized edge bundling technique based on edge advection in compatiblevector subspaces, which extends the recent kernel density estimationedge-bundling (KDEEB) [23] with the following contributions:

1. generation of bundled layouts where edge compatibility is de-fined by one or several numerical edge attributes, e.g., direc-tion, time, or weight;

2. a simple and scalable GPU implementation of our method;3. applications of ADEB for exploring trail (airline trajectory and

eye tracking) datasets demonstrating the method’s added valueas opposed to classical bundling of undirected graphs.

The structure of this paper is as follows. Section 2 reviews relatedwork on edge bundling and trail visualization. Section 3 presentsADEB’s mathematical framework and its implementation. Section 4applies ADEB to the visual analysis of traffic and eye tracking data.Section 5 discusses our method. Section 6 concludes the paper.

∗e-mail: vsevolod.peysakhovich@isae.fr†e-mail: christophe.hurter@enac.fr‡e-mail: a.c.telea@rug.nl

2 RELATED WORK

We next briefly review several state-of-the-art edge bundling meth-ods and trail visualization techniques.

2.1 Edge bundling techniques

Key to edge bundling is the reduction of edge-edge clutter by group-ing so-called compatible edges into spatially compact and thin bun-dles. This allows an easier visual detection of coarse-scale con-nectivity patterns in the input graph and supports tasks like findingnode-groups which are connected to each other by edge-groups (bun-dles) [13, 41]. Such strategies are similar to cartographic map gener-alization, concerned with legibly depicting a complex world in static2D views [5]. Clutter causes and reduction strategies are discussedby Ellis and Dix [11] and Zhou [45]. Edge bundling has been of in-creased interest in recent research. We group them according to theedge simplification level (data, geometry, and image), as follows.Data-based methods: Such methods reduce clutter by showing asimpler graph obtained by filtering or by aggregating edges [11]. In-teractive tools, such as NodeTrix, compact dense subgraphs into ma-trix representations [16, 17]. Ploceus displays networks from dif-ferent perspectives, at different abstraction levels, and with differentedge semantics [33]. Filtering can be done by user-specified queriesbased on multivariate criteria, or by automated methods using edgemetrics such as centrality [43] or spanning trees [32].Geometry-based methods: Large graphs can be visualized withlimited clutter by edge bundling techniques. Edges found to be com-patible, i.e. close in the graph drawing and (optionally) similar inattribute values are deformed to overlap, thereby increasing separat-ing whitespace and thus reducing clutter at the expense of increas-ing overdraw. Dickerson et al. merge edges by reducing non-planargraphs to planar ones [8]. Flow maps use a binary clustering of graphnodes to route curved edges along [36]. Several authors use flow mapcontrol-meshes to cluster and route curved edges [38, 46]. Holtengeneralized this idea by routing edges of compound graphs along thegraph’s hierarchy drawing [19]. Gansner and Koren bundle edges ina circular node layout by area optimization metrics [14]. Edge rout-ing can also use Delaunay [7] and Voronoi diagrams [28, 29] reflect-ing the node positions. The most popular (and versatile) bundlingmethods use force-based techniques that use a force field designed toequal the gradient of a quality function to optimize [10, 20], and havebeen adapted to separate bundles running in opposite directions [40].A major issue of geometric techniques is computational complexity.To reduce this, graph optimization techniques and multilevel cluster-ing numerical methods have been proposed [13, 37].Image-based methods: These methods exploit the parallel com-puting power of modern GPUs to speed up bundling and/or offernew visual metaphors for bundled graphs. Visual techniques includecolor interpolation to show edge directions [7, 19]; and mapping at-tributes such as local edge density or edge lengths via transparencyor hue maps [22, 26, 29, 40]. Bundles can be drawn as compactshapes emphasized by shaded cushions [39, 41]. Animated texturescan be used to convey edge directions for both static and dynamicgraphs [24]. Image-based methods can also massively acceleratebundling: Skeleton-based edge bundling (SBEB) creates stronglyramified bundles by using GPU-computed skeletons or medial axesof the graph drawing’s thresholded distance-transform as bundling

cues [12]. Recently, the force-directed idea of [20] was adapted fora GPU setting, offering speed-ups of up to two orders of magnitudecompared to state-of-the-art geometric bundling methods [23, 24].

2.2 Trail visualization

Trail datasets consist of sequences of points recording the motion ofseveral objects, e.g. vehicles, or tracked trajectories of object parts,e.g. eyes or human silhouettes. Such datasets have three main fea-tures. First, they can be described as attributed graphs, where nodesrepresent trajectory endpoints and (curved) edge control points repre-sent actual motion paths. Attributes include motion direction, speed,and timestamps. Secondly, they are typically large: For instance,15 minutes of eye tracking data recorded at 50 Hz already gives45K sample points; air traffic recorded at a country’s scale over aweek can yield over a million sample points [22]. Finally, they arecomplex, e.g. contain many intersecting trails covering large spatialdomains. All these factors make trail exploration a challenging pro-posal with respect to attribute depiction, clutter reduction, and com-putational scalability. We discuss below several methods related tothis task.Trajectory exploration: Interactive brushing-and-linking tech-niques help locally reducing clutter when exploring large multivari-ate trajectory datasets, by selecting up to three dimensions to showin a single 2D view [26]. Focus+context interaction helps furtherreducing clutter and posing complex spatial-and-data queries [25].Extracting events of interest from different kinds of spatio-temporaldata is discussed (but not fully implemented) by Beard et al. [3]. Thesame query idea is used to extract trajectory parts in [2, 15].Eye tracking exploration: Density maps tackle the visual scalabil-ity problem by aggregating spatially close information for eye trackanalysis [1, 2, 34]. Separately, Burch et al. transform eye trackingdata into a dynamic graph to help exploration [4]. Areas of interest(AOI) analysis provides information about dwell times, transitions,fixation points, and AOI hits, but require accurate a priori knowledgeto define such areas [9, 18]. A few methods exist that are able togroup eye-movement paths; however, they are neither visual nor in-tuitive to use [31]. An overview of eye movement pattern analysis isgiven by Laube [30].

Given the size and clutter of trail datasets, bundling methods haveemerged as an effective way to show the coarse-scale connectivitystructure of these datasets [7, 12, 20, 22, 23, 29]. However, as analyz-ing trail datasets requires reasoning about trail attributes (e.g. time-stamps, speed, and direction), undirected bundling techniques are oflimited use. For instance, a bundle connecting two node-groups Aand B will only tell that A and B are connected, but not in whichdirection(s) and at which time instant(s) objects moved from A to B.As a side effect, mixing edges of different attribute values in a bundlemakes the attribute-based shading-and-blending schemes proposedby several authors [20, 22, 29] of limited effectiveness. Conversely,using bundling techniques for attributed graphs [20, 40] is not prac-tical, as these are far too slow for large datasets such as trail data.

3 ADEB FRAMEWORK

We next describe our ADEB method, a framework that bundles largegraphs using an edge-compatibility scheme that can be flexibly con-trolled to include (mixes of) edge attributes. We first outline the mainelements of the KDEEB algorithm from which ADEB is inspired(Sec. 3.1). Section 3.2 presents ADEB’s mathematical details. Sec-tions 3.3 and 3.4 explain how to use attributes to create various edgecompatibility metrics. Implementation details are given in Sec. 3.5.

3.1 Kernel density bundling

We first review the main principles of the kernel density estimationedge bundling (KDEEB) technique. Let G = ei1≤i≤N ⊂ R2 be thedrawing of a graph of N edges, where each edge ei is represented

by a straight line or arbitrary 2D curve. First, an edge-density mapρ : R2→ R+ is computed using kernel density estimation

ρ(x ∈ R2) =1h2

N

∑i=1

∫y∈ei

K(

x−yh

), (1)

where K : R2→R+ is a bivariate radial kernel obtained from a sym-metric univariate non-negative kernel function of bandwidth h > 0,e.g. Gaussian or Epanechnikov. The bundling of G is given by thefixed point of the ordinary differential equation

dxdt

=h(t)∇ρ(x, t)

max(‖∇ρ(x, t)‖,ε), ∀x ∈ G, (2)

with initial conditions given by the input graph drawing G. Theregularization constant ε 1 takes care of zero gradients. Aftera few Euler iterations used to solve Eqn. 2, during which one re-samples edges, recomputes ρ , decreases h, and makes a 1D Lapla-cian smoothing pass over G’s edges, KDEEB converges and aggre-gates edges into bundles. Note that KDEEB is nothing else but thewell-known mean-shift algorithm [6] applied to the graph drawingG. Hence, KDEEB inherits smoothness, noise robustness, and con-vergence results proven for mean-shift. For details, we refer to [23].

3.2 Mathematical model for ADEBWe proceed by first introducing a few necessary definitions.Edge directions: Consider a given drawing G = ei1≤i≤N of a di-rected graph or trail dataset. For each point x ∈ ei ⊂ G, we denoteby d(x) the unit vector tangent to the curve representing ei, orientedin the direction given by walking on ei from its start to its end point.For our trail datasets (Sec. 2.2), ei will be typically a 2D curve ratherthan straight line (as in a classical node-link graph drawing), so dchanges along ei. Secondly, note that d encodes the local directionsof the initial graph drawing, given as input to the bundling process.Flow direction map: Given a graph drawing G as above, the flowdirection map θ : R2→ R2 is defined as

θ(x ∈ R2) =1h2

N

∑i=1

∫y∈ei

d(y)K(

x−yh

), (3)

where K is the same kernel as in Eqn. 1. We can think of θ as beinga 2D viscous fluid flow whose boundary conditions are given by thed values on G. The bandwidth h in this analogy is the fluid viscosity.Note that several points have θ(x) = 0, e.g., points equidistant fromtwo parallel opposite-direction edges of. In our analogy, these wouldbe flow separation lines. Points x close to an edge have values θ(x)close to the edge direction d. The flow magnitude ‖θ‖ decreasesmonotonically and smoothly away from the flow boundary G.Directional compatibility: At each point x ∈ G, we define a sub-space Ωx,c of compatible directions as

Ωx,c =

y ∈ R2 \ker(θ)

∣∣∣∣d(x) ·θ(y)‖θ(y)‖≥ c⊂ R2, (4)

where ker(θ)= x∈R2|‖θ(x)‖= 0 and c∈ [−1,1] is a compatibil-ity factor. This factor represents the cosine of the maximum allowedangle between the edge direction and flow direction at y. Thus, Ωx,0covers all points where the flow direction deviates from d(x) by π/2at most, and Ωx,−1 ≡ R2. In general, given a point x of the initialgraph drawing G, Ωx,c gives all points in R2 where the flow map θ

has roughly the same direction as x, subject to the factor c.Given the ρ and θ maps, the subspace Ωx,c is defined at each point

x∈G. We can now define the ADEB bundling of G as the fixed pointof the ordinary differential equation

dxdt

=h(t)∇Ωx,c ρ(x, t)

max(‖∇Ωx,c ρ(x, t)‖,ε), ∀x ∈ G (5)

In contrast to KDEEB (Eqn. 2), the gradient ∇Ωx,c ρ is estimated nowonly over the subspace of compatible directions Ωx,c. In analogywith KDEEB, we solve Eqn. 5 by a few Euler iterations, duringwhich we resample edges, recompute ρ and θ , decrease h, and makea 1D Laplacian smoothing pass to remove small wiggles.

AD

EB

(c=

co

s π

)A

DE

B (c=

co

s π/6

)In

pu

t d

ata

se

ts

Figure 1: Influence of compatibility factor on bundling result.

In summary, while KDEEB advects points of a given drawing to-wards local maxima of its edge density, ADEB advects points to-wards local density maxima of compatible directions. Figure 1 showsthe influence of the compatibility factor c on four simple graphs.Edges are colored using a directional colormap (see color wheel infigure: edges going west are red, edges going north are purple, edgesgoing south are green, and edges going east are blue). Luminanceis linearly increased along edges, so their direction is also shownby following the dark-to-saturated color gradient, in line with [21].The top row shows the input graphs, which are simple for illustrationpurposes. The middle row uses c = cos(π), in which case ADEBproduces identical results with KDEEB. As shown by the colors,bundles contain edges running in different directions. This createsbundles which arguably do not reflect well the connectivity patternin the input graph, see e.g. the second row from left. The bottomrow shows ADEB for c = cos(π/6), which creates bundles contain-ing only edges having similar directions. As argued also by Selassieet al., this is useful to separate a graph into several bundled ‘layers’,each representing the bundling of a subset of edges having locallysimilar directions [40]. The bottom-right image in Fig. 1 shows this:The input graph (top-right image), which contains edges of alternat-ing directions as indicated by the arrows, has been effectively bun-dled to yield two similar red and purple bundle structures, one foreach input direction (compare this example with Fig. 7 in [40]).

3.3 Attribute-based bundling

Apart from direction, we can easily use other edge attributes to defineedge compatibility. Consider for instance that edge ei has, at pointx, an attribute value ti(x) ∈ R. We next transform ti to a vector ti =(tx

i , tyi ) ∈ R2 by using polar coordinates, i.e. set

txi (x) = cos

(ti(x)− tmin

tmax− tmin·π), ty

i (x) = sin(

ti(x)− tmin

tmax− tmin·π)

(6)

with ti ∈ [tmin, tmax] over the entire G. Next, we use ADEB (Sec. 3.2)unchanged. The resulting flow direction map encodes the compat-ibility of our edge-attribute ti. The compatibility factor c is to beset accordingly: For instance, if edges are compatible in time whentheir timestamp difference is less than σ seconds and the maximumtimestamp value is tmax, then c should be set to cos( σπ

tmax).

3.4 Multi-criteria bundling

We can use the ADEB framework to bundle graphs using compati-bility criteria defined by several attributes. Suppose that we calcu-lated two flow direction maps corresponding to edge direction andedge timestamps respectively. We can next compute two subspacesof compatible directions and compatible timestamps. Next, by com-puting the gradient ∇ρ over the intersection of these two subspaces,the result given by Eqn. 5 is a multi-criteria bundled layout.

Figure 2 shows this for a dataset recording the gaze for a one-minute exploration of Ilya Repin’s painting Unexpected Visitor. Sim-ilar to Yarbus [44], a subject was asked to look at the painting andfirst assess and verbalize the age of the main characters, and next theage of secondary ones. Using time-based compatibility, ADEB suc-cessfully bundles together the trails separated in time (Fig. 2 b, trailscolored by time via a blue-green-red colormap). We find that thereare two main ‘visual scans’ in this scenario – first, a scan betweenthe faces of foreground characters A and B (green bundle), followedby a second scan between background characters C and D (red bun-dle). However, since we don’t use direction for bundling, we cannottell if in these scans the eye moved from left to right, right to left, orboth directions. Figure 2 b shows the result of directional bundling,colored by trail directions like in Fig. 1. We now see that both scansA-B and C-D include left-to-right (blue) and right-to-left (red) bun-dles of similar thickness. Hence, the eye moved in both cases in bothdirections, and it did so in a balanced way (comparable amount oftimes in both directions). However, close to character C, the ends ofthe C-D bundles are pulled towards and mixed with the A-B bundles,even though the two scans are temporally separated. Using time-and-direction based compatibility, we can obtain both desirable effectsshown before (Fig. 2 d): Each scan is separated in a left-to-right andright-to-left bundle, and temporally different scans are not mixed.

3.5 Implementation details

We evaluate the two components θx and θy of the flow direction θ

by using Eqn. 3 twice, for the two corresponding components dx anddy of the edge direction vector d respectively. We implement this onthe GPU by splatting the kernel K at all edge sample points usingadditive blending, and accumulating the resulting θx and θy values intwo floating-point textures. Next, we compute the density gradient∇Ωx,c ρ by finite differences, on a compatible subspace Ωx,c ∩νε (x),where νε (x) is a neighborhood of x of radius ε ≥ h> 0. While cover-ing the neighborhood νε (x), each point y of it is tested for directionalcompatibility. The finite differences are computed only on pointsthat pass the compatibility test in Eqn. 4. Finally, while doing thebundling iterations that solve Eqn. 5, we relax the user-given com-patibility value c so it reaches cosπ at the last iteration. This way,the first iterations force edges to aggregate into compatible bundles,while the last iterations ‘compact’ edges already found to be com-patible into smoother final bundles.

Our method has similar complexity and computational perfor-mance with KDEEB. Specifically, computing the flow direction mapθ takes twice the time required for KDEEB’s density map com-putation. Estimating the directional compatibility criterion addsonly a simple logical condition for the density gradient computa-tion. Memory-wise, our method requires two floating-point buffersof the size of the graph drawing G, which is a negligible cost in-crease as compared to KDEEB. Compared to FDEB [20] and dividededges [40], two other prominent directional bundling techniques, ourmethod is over one order of magnitude faster (see also Sec. 5).

We render the bundled edges as curves colored by the value ofone attribute of interest, e.g. time or direction. We also scale thecurve thickness at each point x with the local edge-density value ρ(x)(Eqn. 1). Bundles (or bundle fragments) thus appear locally thickerin areas where they contain many edges, thereby enabling users toestimate the importance of the respective connection patterns.

a) Raw eye trails

c) Bundling by direction

b) Bundling by time

d) Bundling by direction and time

AB

C

D

Figure 2: Multi-criteria bundling of an eye tracking dataset.

4 APPLICATIONS

We next present two applications of ADEB for the qualitative analy-sis of trail datasets represented by directed attributed graphs.4.1 Aircraft traffic explorationOur first application concerns the exploration of aircraft traffic.Given a geographical region (e.g., a country), aircraft position is pe-riodically recorded over a given time interval (e.g., days or weeks) bya mix of observations from ground radar stations and signals emittedby onboard transceivers. This generates a set of trails, attributed bythe measured quantities (aircraft timestamp, position, speed, height,and ID). Considering landing and takeoff points as nodes and trailsas edges, such data can be modeled as a directed attributed graph.Such data is used by air traffic controllers both online (to keep safedistances between flying aircraft) and offline (to optimize air trafficflows, e.g., minimize delays and optimize flight routes).

Edge bundling has already been used to highlight coarse-scaleconnection patterns in air traffic data [23, 22]. However, this causesproblems when tasks at hand involve reasoning about flight direc-tions. Indeed, existing directional bundling techniques are either notscalable enough [40] or need expensive preprocessing steps [12]. Toseparate different directions mixed in the same bundle, interactioncan be used, e.g. the MoleView semantic lens that smoothly inter-polates between bundled and unbundled trails at locations of inter-

est [25]. However, this approach only provides local insights.Dataset: The analyzed dataset is 24 hours (February 8th, 2009)of recorded aircraft trails over France (Fig. 3, 18858 trajectories,415257 records or sample points). We detail below two explorationscenarios, performed by an ATC expert, using our ADEB bundlingtechnique, embedded in the FromDaDy visual exploration tool [26].Findings have been cross-checked by a separate air traffic controller.Overview analysis: Exploration started by analyzing an overviewof the flight patterns. Figure 3 a shows the input trail set. Trails arecolor-coded based on direction, as in Fig. 2. To emphasize direction,we made them dark and saturated at the start points and bright anddesaturated at end points. Figure 3 b shows the directionally bundledtrails, color-coded as above. Compared to the unbundled image, wecan now see the different incoming and outgoing aircraft ‘flows’ atseveral main European cities (Paris, London, Zurich). We’ll explorethese further below. Secondly, we notice the complex flight patternover Lyon, which differs significantly from the pattern above Paris.Indeed, while Paris does not have transit traffic (aircrafts that fly overa given area without landing), Lyon has major transit flows. Onesuch example is formed by the red and blue bundles linking Zurichto Spain (Fig. 3 b, bundle b). Over the entire image, we see sev-eral such bundle-pairs having similar thickness in the two directions,e.g. the one ending in London (Fig. 3 b, bundle a). Since bundlewidth maps the number of bundled trails (Sec. 3.5), this shows usthat such paired flows contain balanced traffic in the two directions.Such flows correspond to daily commuting flights.Detailed analysis: We next zoom to explore flight details in a circleof about 100 km around the Paris area (Fig. 3 c). We discover thatthis flow pattern has four main incoming and four outgoing flows(see arrows in the figure). Also, we see that there are only a fewthin bundles whose endpoints are not in the center of the image –thus, we confirm the overview finding that Paris has little transittraffic. Most bundles start/end at two concentrated locations (dot-ted circles in the figure). These correspond to the two main Parisairports, Roissy and Orly. The detail analysis also shows us factsthat were not visible on the overview: For Roissy, the thickness ofthe main bundles (marked by arrows) show that west and north de-partures are far fewer than south and east ones. More interestingly,we see that, at this scale, bundle-pairs corresponding to departure-arrival flows in the same main direction, are far less parallel than inthe coarse-scale visualization. This matches existing knowledge thatlanding-departing patterns close to airports are much wider spread(in direction) than in-flight patterns, as the former need all to sharea smaller geographical area. For Orly airport, we immediately seethat traffic is far smaller than for Roissy, and it goes mainly to thesouth and south-west (bundles sw and s). Only a thin bundle goesfrom Orly to the north (arrow n). This matches known evidence onthis airport: Orly mainly serves domestic flights and is in the north ofFrance; as such, its traffic goes mainly to the south and south-west.

Figure 4: Aircraft traffic over Paris (bundled with KDEEB). Com-pare with the ADEB directional bundling in Fig. 3 c.

a b c

Paris Zurich

London

Spain

Figure 3: Aircraft trails analysis. (a) Raw data. (b) Directional bundling over France. (c) Zoom-in over Paris area.

Such insights cannot be obtained with a direction-agnosticbundling method. To illustrate this, Fig. 4 shows the same zoom-in asin Fig. 3 c, bundled with KDEEB. The north departure bundle is nowmixed with north-east arrivals, and east departures get mixed withsouth-east arrivals. Also, the several thin traffic bundles (aircraft thatdo not start/stop at Paris) visible in Fig. 3 c get now aggregated. Inother words, undirected bundling keeps the very-coarse connectionpatterns, but eliminates the finer ones that depend on direction.

4.2 Eye tracking explorationRecent eye-tracking technology developments made eye movementdata increasingly used in applications in neuroscience, psychology,human factors, training, and marketing. Such data consists of sam-pled trails giving the motion of the subject’s gaze while completinga given task. Its two most important elements are fixations and sac-cades. Fixations describe points that the eye is attracted to (thus,where attention is drawn). Saccades link fixations and model howone ‘scans’ the image to link information given by fixations.

For fixation events, studied more often, attention and saliencymaps are often used to show regions of interest by locating high-density fixation areas. However, the saccades linking such points arealso of great importance, especially to understand how low-level fix-ation information is aggregated to form a higher-level understandingof the picture. Specifically, spatially and temporally close saccadesare to be grouped into so-called scanpaths, and further analyzed tounderstand how the image was processed [44]. Two kinds of visu-alizations exist for scanpaths: either the gaze is dynamically playedback, or the sequence of saccades is depicted statically by connect-ing consecutive fixation points by arrows. Both visualizations arequite inconvenient for pattern analysis. Playback is necessarily lo-cal in both time and space, and thereby cannot show the grouping.Playback is also time-consuming. Static drawings of raw scanpathsgenerate cluttered images, especially when analyzing data producedby lengthy experiments containing thousands of saccades [4, 27].

We next describe the use of attribute-based bundling for analyzingsuch eye tracking datasets. First, we outline how we extracted anattributed graph from raw eye-tracking data. Next, we detail twoexperiments where our bundling proved to be an effective aid forgetting insight into the recorded data (Secs. 4.2.1 and 4.2.2).Preprocessing: Raw gaze data, recorded at 50 Hz, was prepro-cessed to extract fixations and saccades as follows. Consecutivesample points located in a square of 20×20 pixels and separatedby at least 200 ms were considered to be a fixation event. Pointsmarked as a one fixation event were replaced by their average. Thisreduces small-scale noise (scattered close fixation points) generatedby micro-saccades and eye tracking device imprecision during a fixa-tion. Clustering algorithms could be used [6] to further reduce clutterand aggregate adjacent fixations corresponding to one object of in-terest but separated spatially due to the imprecision of human visual

system. Next, trails formed by consecutive sample points were cutbetween fixations, yielding the saccades. Finally, a graph was cre-ated using fixations as nodes and saccades as curved edges. Notethat since human eye does not encode any information during thesaccadic movement, curved edges of our graph do not have directattentional interpretation and thus could be distorted.

4.2.1 Multitask experimentEye tracking technology can be used to study repartition of visualattention. Consider, for example, how a pilot watches the cockpit in-struments during a certain maneuvre (task). Such tasks, e.g. take-offor landing, are typically split into multiple subtasks, each involvingseveral instruments between which the pilot has to allocate attention.For each subtask, instruments have to be watched in appropriate or-der and with a given frequency. For instance, the priority of the Pri-mary Flight Display (PFD) instrument with respect to Flight ControlUnit (FCU) will be different for the final landing approach as com-pared to the cruise phase [35].Data and tasks: This dataset is part of the priority management test-ing of the pilot selection at French Civil Aviation School [35]. Thesubject has to perform a main task containing four concurrent sub-tasks of different nature (tracking, monitoring, target detection, men-tal computing), with two priority conditions: (a) all subtasks haveeven priorities, and (b) two high-priority and two low-priority sub-tasks (uneven priorities). Tasks are performed using a dashboard-likemultitask interface on which several instruments show dynamically-changing data. The interface and location of the various instrumentsused for each task is shown in the background of Fig. 5a. The top barwidgets let subjects monitor their performance (four bars for eachtask, and one aggregated-performance bar). For the uneven condi-tion, subtask priorities are shown in Fig. 5b. A detailed descriptionof this experiment and the multitask design are given in [35]. Twosubject groups, one formed by experienced pilots, and the other oneformed by novices, performed the task under both priority condi-tions, while their gaze was continuously recorded. For each suchrun, which took 4 minutes, we used the described above preprocess-ing to obtain a graph having 234 saccades of 24 samples each onaverage. Using this data, we next try to find similarities and differ-ences in task execution across users and priority conditions.Results: Figure 5 shows the result of applying directional bundlingon four sample test runs, involving both priority conditions anduser groups, colored by direction as in earlier figures. Directionalbundling allows us to identify the main scanpaths (characteristic se-quences of saccades) involved in each run, and also to compare thesescanpaths across users and conditions. Data is normalized in thesense that all runs are of the same length (in time and number ofsample points), and also contain similar numbers of fixation points.As such, differences between such bundled images reveal both inter-subject and intra-subject similarities and differences.

Tracking

Monitoring

Target detection

Computing

Taskperformance

bars

Aggregatedperformancebar

(a) Novice user performance with even priorities

high priority

high priority

low priority

low priority

(b) Novice user performance with uneven priorities

1

2

3

34

4

(c) Expert performance with even priorities

1

1

1

(d) Expert performance with uneven priorities

Figure 5: Bundled eye-tracking trails of novice and expert subjects for a multitask experiment done with two priority conditions.

Let us first analyze the influence of expertise level on the visualattention distribution. The expert has more pronounced transitionsbetween the four subtasks as compared to the novice (Figs. 5a, 5b), asshown by the thicker bundles in Figs. 5c, 5d. Given that the averagesaccade speed is similar over humans, and the experiment durationis identical for both user types, we deduce that the expert needs lesstime to perform a subtask and spends more time to switch betweenthe four screen areas to update his knowledge about subtasks. Healso largely ignores the performance bar in his ‘attentional walk’ ascompared to a novice. This way of scanning an image, also calledocular strategy, leads to better performance [35].

The ocular strategy changes when the priorities of the four sub-tasks are uneven, as shown by the different structure of the bundlesin Figs. 5b,5d as compared to Figs. 5a,5c. Yet, the priority conditionaffects novices and experts differently. The expert shifts his attentionto the more important subtasks, as shown by the increased bundlecoverage in the right part of Fig. 5d. In contrast, the novice almostabandons the monitoring task and spends most of the time on the twohigh-priority subtasks. For the uneven priority case, the tracking taskinduces significant attention for both the expert and novice. This canbe due to the fact that, in this experiment, the user’s right hand al-ways rests on the joystick that controls the tracking subtask, so theyunconsciously dedicate significant attention to this task, even whenthey are instructed that tracking is a low-priority subtask.

Directional bundling also shows us the sequence (order) of atten-

dance of different instruments. Consider the expert participant witheven priorities (Fig. 5c). The main transitions between subtasks, cov-ering about 80% of the entire set of saccades, are captured by thefollowing thick bundles: tracking to monitoring (green bundle 1),monitoring to computing (two merging blue bundles 2), computingto target detection (purple bundle 3), computing to tracking (red bun-dle 4). Hence, a special order of subtask monitoring can be deduced.First, we deduce that the expert monitored information in anticlock-wise order (tracking, then monitoring, then computing, then track-ing again). We also see, for the even priority case, that neither theexpert nor the novice do have significant transitions between targetdetection and monitoring. For the uneven priority case, however,the expert has many transitions from monitoring to target detection(Fig. 5d, blue bundles 1). This shows that the expert rememberedthat target detection is of high priority, and periodically switches at-tention to it from the less important monitoring.

4.2.2 Landing experiment

Data and tasks: This dataset is part of a study conducted at theFrench Aerospace Engineering School. During the experiment, apilot performed a landing manoeuvre in a flight simulator. The ex-periment lasted 15 minutes. The aim was to test a new cockpit instru-ment providing landing assistance, which we next call the LandingAid Instrument (LAI). More specifically, we wanted to understandwhether (and how) the new LAI is used along the other instruments.

Eye tracking recording resulted in a graph with 1194 saccades of 8samples each on average.

Results: Figure 6 shows the raw trails from this experiment, theresults of our ADEB bundling, and those of undirected (KDEEB)bundling. Bundles are directionally colored, as in earlier figures. Thebackground image shows the cockpit dashboard. Carefully inspect-ing the raw trails (Fig. 6 a) reveals several saccades connecting thePrimary Flight Display (PFD, left), Navigation Display (ND, center),Flight Control Unit (FCU, top-right), and the new Landing Aids In-strument (LAI, bottom-right). However, the raw trails display is toocluttered. We cannot be sure that we found all relevant instrument-to-instrument trails, and we cannot see high-level scanpath patterns.When using ADEB bundling (Fig. 6 b), the following connectionsbecome obvious: faint purple bundle 1 LAI→FCU, blue bundle 2PFD→LAI passing by ND, red bundle 3 in the opposite direction,red bundle 4 FCU→PFD, and blue bundle 5 in the opposite direc-tion. The bundle PFD-ND is denser than the PFD-FCU one, andmuch denser than the PFD-LAI one. Thus, the main pattern in thepilot’s gaze can be deduced: PFD→ND→FCU→PFD. The LAI israrely consulted because of its novelty – the pilot is used to land theaircraft without this instrument. In more detail, we see that thereis no dense connection FCU→LAI. The pilots explained that, afterentering flight parameters on the FCU, they are used to check themimmediately on the PFD (as shown by the red bundle 4). Thus, tobetter integrate the new LAI in this maneuvre, a new training proto-col should be designed in which pilots are specifically told when toadd the visual consultation of the LAI in their action sequence.

Figure 6 c shows the results of undirected bundling. We see,as in our earlier aircraft trail-analysis (Fig. 3 c vs Fig. 4), that undi-rected bundling highlights only coarse-level connection patterns andsmooths our finer-grained ones. For example, the red LAI→PFD andFCU→PFD bundles visible with ADEB (Fig. 6 b) are now hard tosee, as they appear to end at ND. More problematically, we notice aquite thick bundle between FCU and LAI (Fig. 6 c, bundle 1), whichsuggests that the subject did in fact frequently use the LAI in con-junction with the FCU, which we know is not true. ADEB bundlingdoes not show this bundle, as there are not enough saccades coherentin both space and time between FCU and LAI, and thus conveys usthe correct insight (LAI rarely used).

5 DISCUSSION

We discuss next several technical aspects of our method.

Generality: Most earlier bundling methods use only edge length,absolute angles, and relative distances (the latter which we alsouse). So far, only [40] explicitly shows directional bundling; [20]mentions this possibility, but does not actually demonstrate it.Separately, attribute-based bundling is noted as a possibility in [41],but not actually demonstrated. In contrast to all above, our ADEBdemonstrates how both local edge direction and edge attributes canbe added to earlier geometric compatibility metrics.

Scalability: The most prominent bundling techniques using edgecompatibility measures (beyond edge distance) are [20, 40]. Bothhave a complexity of O(E2C) where E is the number of edges in theinput graph, and C the number of edge sampling-points. In contrast,our method, just as KDEEB [23], is O(C). This makes our methodconsiderably more scalable than [20, 40], and of about the samescalability as [23]. In detail, FDEB takes 19 seconds on a graph of2101 edges and 205 nodes (the well-known US airlines dataset),with 12 sample-points/edge on average (thus, roughly 25K samplepoints in total); the method in [40] takes 24 seconds for the samegraph, for a sampling of up to 25 points/edge (up to 75K samplepoints), excluding preprocessing costs which authors note to be 10%of the total cost. For the same graph, KDEEB takes 0.5 seconds, at86K sample points. ADEB takes 1.3 seconds for the same amountof sampling points. Overall, ADEB is roughly twice as slow as the

x

a) Raw eye trails

c) KDEEB bundling

b) Directional bundling

PFDND

FCU

LAI

1

2

5

3

4

1

Figure 6: Raw and bundled eye trails, landing scenario (Sec. 4.2.2).

undirected KDEEB, but about 18 times faster than the directionalmethod in [40].

Interaction: Edge bundling aims to create simplified overviewsof complex graphs. To solve specific tasks, extra tools suche.g. local levels-of-detail [25], attribute-based selection [26], orinteractive aggregation [42], are obviously beneficial. Such toolscan be easily added to ADEB. In line with earlier bundling pa-pers [7, 12, 20, 23, 29, 40], we do not detail such extensions here,so that our contribution on directional bundling is more focused andeasier to separate from such additional tooling mechanisms.

Limitations: Our attribute-based bundling implies that we can mapdistances between attribute-values (in their original space) to angledistances (Eqn. 6). This is immediate for quantitative and ordinalattributes, but is not evident for categorical attributes. For the lat-ter case, such a mapping would require an algebra definition on thecategory space. Hence, for categorical attributes, we suggest to clus-ter trails according to their category and perform separate bundlingcomputations for each such cluster (much like [41]). Separately, weshowed the added-value of ADEB vs a single undirected bundlingmethod (KDEEB). This leaves the open question whether other undi-rected bundling methods would be comparably better. While such

comparisons are yet to be done, we argue that, for tasks that requireseeing and analyzing bundles of different directions, all undirected-bundling methods would exhibit similar limitations as KDEEB. Fi-nally, we note that our use-cases of ADEB for analyzing eye-trackingdata do not imply that ADEB is the ultimate technique to gain alltypes of insights from such data, as compared to e.g. areas of inter-est (AOIs), dwell-time-per-AOI, or transition matrices. Rather, weemploy this use-case to show how ADEB can be of added value asopposed to a raw or undirected-bundling display of trails (Fig. 6).6 CONCLUSION

In this paper, we proposed ADEB, a method that creates edge bun-dles from attributed graphs where edge compatibility can be definedby one or several attributes. ADEB is fast, simple to implement,scalable, generic, and could be easily extended to handle dynamicgraphs. We demonstrated ADEB, and implicitly the added-value ofdirectional bundling, by showing two applications in the analysis ofaircraft trails and eye movement recordings.

Future work can target several aspects. Refined compatibility met-rics (based on graph attributes) and visual metaphors could allowusers to better spot specific patterns of interest in large eye-trackingdatasets. Secondly, ADEB could be adapted to target additional do-mains, such as creating simplified attribute-based visual representa-tions of DTI fiber tracts. Finally, we aim to explore how ADEB canbe extended to handle attribute-based bundling of dynamic graphs.

ACKNOWLEDGEMENTS

The authors thank Nadine Matton, Frederic Dehais and SebastienScannella for providing gaze datasets.

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